REFLECTIONS

CONSIDER THE FIRST QUADRANT point (a, b), and let us reflect it about the y-axis. It is reflected to the second quadrant point (−a, b).

If we reflect (a, b) about the x-axis, then it is reflected to the fourth quadrant point (a, −b).

Finally, if we reflect (a, b) through the origin, then it is reflected to the third quadrant point (−a, −b). The distance from the origin to (a, b) is equal to the distance from the origin to (−a, −b).

Fig. 2 is its reflection about the x-axis. Every point that was above the x-axis gets reflected to below the x-axis. And every point below the x-axis gets reflected above the x-axis. Only the roots, −1 and 3, are invariant.

Again, Fig. 1 is y = f(x). Its reflection about the x-axis is y = −f(x). Every y-value there is the negative of the original f(x).

Fig. 3 is the reflection of Fig. 1 about the y-axis. Every point that was to the right of the origin gets reflected to the left. And every point that was on the left gets reflected to the right. Every x becomes −x. Only the y-intercept is invariant. The equation of the reflection of f(x) about the y-axis is y = f(−x).